A NONPARAMETRIC SYSTEM IDENTIFICATION BASED ON TRANSIENT ANALYSIS WITH PLANT PROCESS OF HEAT EXCHANGER AS STUDY CASE

In this article, a nonparametric identification system based on transient analysis has been reviewed, by taking the case in some of the data plant process of heat exchanger. Results of the study found that the first-order transfer function without time-delay the proposed model to the data with a temperature constant value is 35.20 oC and the time constant is 7200 seconds. This model has been fit to meet the existing data proving that the results of the calculation error do not exceed 2%.


Introduction
Introducing a mathematical model to represent the actual system is very important especially in simulation and prediction purposes, particularly for designation of digital control and dynamic systems.Basically, there are two ways of constructing mathematical models i.e. mathematical modeling and system identification [1].A mathematical modeling is an analytical approach.For example, basic laws from physics (such as Newton's laws and balance equations) are used to describe the dynamic behavior of a phenomenon or a process.On the other hand, system identification is an experimental approach where a model is then fitted to the recorded data by assigning suitable numerical values to its parameters.In many cases such as plant processes are so complex that it is not possible to obtain reasonable models using only analytical approach (physical insight, e.g.balance equations).
In such cases one is forced to use system identification techniques.It often happens that a model that is based on physical insight contains a number of known parameters even if the structure is derived from physical laws.On the other hand system identification methods can be applied to estimate the unknown parameters.To a dynamic system, the system identification is a technique to estimate the mathematical models based on data observed from the system [2,3].Many researchers such as [4], [5], and [6] are identifying the dynamic model of heat exchanger using system identification.
Based on [1], techniques used in system identification can be divided into two distinct methods, i.e. parametric and nonparametric methods.The parametric method is built through a model structure determination which is described by a set of parameters while the nonparametric method is determined based on an output system response in a function or a graph forms.In general, a parametric method can be characterized as a mapping from the recorded data to the estimated parameter vector.
A typical example of the nonparametric method is transient analysis where the input can be a recorded based on step signal and step response.This response will by itself give certain characteristics (dominating time constant, damping factor, static gain, etc.) of a process.Nonparametric techniques are often sensitive to noise and do not give very accurate results.However, as they are easy to apply they often become useful means of deriving preliminary or crude models.Therefore, in order to solve the difficulties to get the dynamics model of process system, one can use system identification.

Nonparametric Identification
This section describes a nonparametric system identification based on a transient analysis.A transient is defined if an input is taken as a step or impulse, and the recorded output constitutes the model [1].Such identification methods are characterized by the property that the resulting models form curves or functions, which are not necessarily parameterized by a finite-dimensional parameter vector.Sometimes, it is of interest to fit a simple low-order model to a step response.This is illustrated as the first-and the second-order systems, which are described using the transfer function model in Eq. ( 1) where Y(s) is the Laplace transform of the output signal y(t), U(s) is the Laplace transform of the input signal u(t), and G(s) is the transfer function of the system [1].
(1) Consider a transfer function system which represented by Eq. ( 2).In such a case, the system is therefore described by a first-order differential Eq. (3).Note that a time delay τ is included in the model.The step response of such a system is illustrated in Fig. 1 that demonstrates a graphical method for determining the parameters K, T and τ from the step response.The gained K is given by the final value.By fitting the steepest tangent, T and τ can be obtained.The slope of this tangent is K/T, where T is the time constant.The tangent crosses the t axis at t=τ, which is the time delay [1]. (2) (3) Fig. 1 Step response first order system with time delay Physically, this equation describes a damped oscillator.After some calculations, the step response is found to be as expressed by Eq. ( 4).The output response, y(t), in Eq. ( 4) for various ζ is illustrated by Fig. 2 [1].
(4) Fig. 2 Step response damped oscillator Fig. 2 obviously shows that the relative damping ζ influences the character of the step responses.The remaining two parameters, K and , merely act as scale factors.The gained K scale is the amplitude axis while scale is the time axis.The three parameters of the model in Eq. ( 2), namely , ζ and could be determined by comparing the measured step response with Fig. 2 and choosing the curve that is most similar to the recorded data.However, one can also proceed in a number of alternative ways.One possibility is to look at the local extreme (maxima and minima) of the step response.With some calculation it can be found from Eq. ( 4).At times, it occurs as given by Eq. ( 5).Where Eq. ( 6) defines time and overshoots is given by Eq. ( 7) [1].
(5) (6) (7) The relationship between the overshoot and the relative damping is illustrated in Fig. 3.The parameters , ζ and can be determined based on values obtained in Fig. 4. The gained is easily obtained as the final value once convergence is achieved.The overshoot has been determined in several ways.One possibility is to use the first maximum.An alternative is to use several extreme and the fact as given by Eq. ( 6).The amplitude of the oscillations in Eq. ( 5) is reduced by a factor for every half-period.Once is determined; ζ can be derived from Eq. ( 7), and will produce Eq. ( 8) [1].From the step response, the period of the oscillations has also been determined.From ( 6), the period is given by Eq. ( 9) [1].Then is given by Eq. ( 10) [1].

Nonparametric Model
A transfer function of shell and tube heat exchanger is described as Eqs.(11) up to (17).It has complex forms and it is difficult to apply to the design of practical controllers.In some cases, it is found that the outlet responses are well approximated by using the first-or second-order system to replace the complicated forms of the transfer functions obtained directly from the transformed solutions.This transfer functions are a model form based on the Laplace transform and it is very useful in analysis and design of a linear dynamic shell and tube heat exchanger.
To obtain a nonparametric model, a step response analysis is used in this case.The dynamic model in Eq. ( 11) can be simplified as Eq. ( 12).The obtained transfer function which assumes is shown in Eq. ( 13) where is the transfer function, is the output signal from the cold water temperature outlet in shell side heat exchanger, , and is the input signal from the hot water flow rate inlet in tube side heat exchanger, .The first order transfer function with time delay is given by Eq. ( 19), where is the gain, is the time constant and is time delay.Fig. 6 Step response first order transfer function with time delay A transfer function of a time delay which is given by Eq. ( 20) is irrational transfer function (it has no s-polynomial in the numerator and in the denominator).In some situations as in frequency response based analysis of control systems containing a time delay, it is necessary to substitute with an approximation in the form of a rational transfer function.The most common approximation is the Padé-approximation as given in Eq. ( 21), where n is the order of the approximation and this order must be chosen [7].The coefficients are functions of n.The Padé-approximations are based on a minimization of the truncation errors in a finite series expansion of .Table 1 shows, as an illustration, the k-values for the orders n = 1 and n = 2 [7].Table 1 shows and which are given by Eqs. ( 22) and (23).Therefore, the Padéapproximation transfer functions for first and second orders are given by Eqs. ( 24) and (25) [7].The step and phase responses of the first order Padé-approximation is shown in Fig. 7. Fig. 8 shows the step and phase responses of the second order Padé-approximation.A second order transfer functions is given by Eq. (28).Where , and refer to system gain, relative damping factor and undamped resonance frequency [1,2].Table 2 shows type of step response for various values of .When and the poles are real and distinct, it is given by Eq. ( 29).

Case Study
The candidate of nonparametric model has been described.When the experimental data obtained are observed, the data are apparently similar to the step response of the first order transfer function.This model is candidate model and has been expressed by Eq. (30).Therefore, the candidate model chosen is the first order transfer function.The error of this model is calculated after offset, where the time of offset is determined according to time constant that occurred.The error is given by Eq. ( 31), where is the error, is the number of the data and is the output signal value with for the data, and for calculated value from the model.The percentage of this error is given by Eq. ( 32), where is within the range of the . ( The step response of the resulted nonparametric model for dataexp1 is shown in Eq. ( 33). (33) The comparison of the model output and the measured dataexp1 is shown in Fig. 9. From Fig. 9, it is obvious that the trend of the model output which is given in Eq. ( 33) is similar to the measured dataexp1.The data and the model start with the same values, i.e.
., and the difference of both temperatures reaches the smallest figure .where is fluctuating between zero to one.These facts are presented in Table 3.   3. The error has been calculated using Eqs.(31) and (32) which takes 8 samples time between 0 and 7000 s.From Table 3, it is obvious that the error is and the percentage of the error is .According to these results, therefore, the model which is shown in Eq. ( 33) is an excellent model because the error is smaller than 2%.In other words, it should be accepted.4. Fig. 11 shows the obtained error.From Fig. 11, it is obvious that the trend of the model output which is given by Eq. ( 34) is similar to the measured dataexp2.The data starts from 29 ℃ and the model also starts from 29 ℃.From time (t) of 0s until to the final time (t) of 7000 s, the difference of the both temperatures reaches the smallest value, i.e. between zeros to one.These facts are presented in Table 4.     From Table 4, it is apparent that the error is and the percentage of the error is .Based on these results, therefore, the model which has been given by Eq. ( 34) is an excellent model because of the error is smaller than .Therefore, it should be accepted.
The step response of the nonparametric model resulted from dataexp3 is shown in Eq. ( 35).The comparison of the model output and the measured data is given in Fig. 13.The error calculation to validate this model is presented in Table 5. Fig. 14 shows the obtained error.   .From the start time of 0s until to the final time of 7000s, the difference of both temperatures reaches the smallest value that it is presented in Table 5. Fig. 14 shows these facts based on the error values in Table 5.The error has been calculated using Eqs.(31) and (32) with 8 samples of time between 0 and 7000 s.From Table 5, it is obvious that the error is and the percentage of the error is .Referring to these results, therefore, the model which is given by Eq. ( 35) is a sufficiently good model because the error is less than .Therefore, it may be accepted.

Fig. 3
Fig. 3 Overshoot M versus relative damping ζ damped oscillator order transfer function without time delay is given by Eq. (18), where is the gain and is the time constant.A step response of such a first order transfer function in Eq. (18) has the following characteristics, which are demonstrated by Fig. 5[7].(18)

Fig.
Fig.8Step and phase's responses order-2 Padéapproximation Fig. 9 Nonparametric model output and data measured dataexp1 Fig. 10 shows these facts based on the error values in Table3.The error has been calculated using Eqs.(31) and (32) which takes 8 samples time between 0 and 7000 s.From Table3, it is obvious that the error is and the percentage of the error is .According to these results, therefore, the model which is shown in Eq. (33) is an excellent Fig. 10 Error calculation nonparametric model dataexp1 Fig. 11 Nonparametric model output and data measured of dataexp2 Fig.12shows the facts based on the values of the error in Table4.The error has been calculated using Eqs.(31) and (32) which takes 8 sample times between 0 and 7000 s.
calculation of nonparametric model and dataexp2 measured

Fig. 12
Fig. 12 Error calculation nonparametric model dataexp2 Fig. 13 Nonparametric model output and data measured of dataexp3

From
Fig.13, it is apparent that the trend of the model output which is given by Eq. (35) is similar to the measured dataexp3.The data and the model start with the same values, i.e.

Table 1 :
Coefficients of Pade-approximations of order and

Table 3 .
Error calculation nonparametric model and data measured dataexp1

Table 4 .
Error calculation nonparametric model and data measured dataexp2

Table 5 .
Error calculation nonparametric model and data measured dataexp3